Graph Structure and Monadic Second-Order Logic: Overview

A popular method for solving reachability in timed automata proceeds by enumerating reachable sets of valuations represented as zones. A naive enumeration of zones does not terminate. Various termination mechanisms have been studied over the years. Coming up with efficient termination mechanisms has been remarkably more challenging when the automaton has diagonal constraints in guards.

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Fast Algorithms for Handling Diagonal Constraints in Timed Automata

Given the versatility of timed automata a huge body of work has evolved that considers extensions of timed automata. One extension that has received a lot of interest is timed automata with a, possibly unbounded, stack, also called pushdown timed automata (PDTA). While different algorithms have been given for reachability in different variants of this model, most of these results are purely theoretical and do not give rise to efficient implementations. One main reason for this is that none of these algorithms (and the implementations that exist) use the so-called zone-based abstraction, but rely either on the region-abstraction or other approaches, which are significantly harder to implement.

/pdf/fast-zone-based-algorithms-for-reachability-in-pushdown-2r3gxm1lb6.pdf

Fast zone-based algorithms for reachability in pushdown timed automata

In this paper, we analyze timed systems with data structures. We start by describing behaviors of timed systems using graphs with timing constraints. Such a graph is called realizable if we can assign time-stamps to nodes or events so that they are consistent with the timing constraints. The logical definability of several graph properties [20], [10] has been a challenging problem, and we show, using a highly nontrivial argument, that the realizability property for collections of graphs with strict timing constraints is logically definable in a class of propositional dynamic logic (EQ-ICPDL), which is strictly contained in MSO. Using this result, we propose a novel, algorithmically efficient and uniform proof technique for the analysis of timed systems enriched with auxiliary data structures, like stacks and queues. Our technique unravels new results (for emptiness checking as well as model checking) for timed systems with richer features than considered so far, while also recovering existing results.

Timed Systems through the Lens of Logic

Reachability relations of timed pushdown automata

A popular method for solving reachability in timed automata proceeds by enumerating reachable sets of valuations represented as zones. A naive enumeration of zones does not terminate. Various termination mechanisms have been studied over the years. Coming up with efficient termination mechanisms has been remarkably more challenging when the automaton has diagonal constraints in guards. 
In this paper, we propose a new termination mechanism for timed automata with diagonal constraints based on a new simulation relation between zones. Experiments with an implementation of this simulation show significant gains over existing methods.

Fast algorithms for handling diagonal constraints in timed automata.

We consider the reachability problem for timed automata having diagonal constraints (like x - y < 5) as guards in transitions. The best algorithms for timed automata proceed by enumerating reachable sets of its configurations, stored in a data structure called "zones". Simulation relations between zones are essential to ensure termination and efficiency. The algorithm employs a simulation test Z <= Z' which ascertains that zone Z does not reach more states than zone Z', and hence further enumeration from Z is not necessary. No effective simulations are known for timed automata containing diagonal constraints as guards. We propose a simulation relation <=_{LU}^d for timed automata with diagonal constraints. On the negative side, we show that deciding Z not <=_{LU}^d Z' is NP-complete. On the positive side, we identify a witness for Z not <=_{LU}^d Z' and propose an algorithm to decide the existence of such a witness using an SMT solver. The shape of the witness reveals that the simulation test is likely to be efficient in practice.

https://drops.dagstuhl.de/opus/volltexte/2018/9566/pdf/LIPIcs-CONCUR-2018-28.pdf/

Paul Gastin

Papers

Fast algorithms for handling diagonal constraints in timed automata.

Reachability in Timed Automata with Diagonal Constraints

Fast Algorithms for Handling Diagonal Constraints in Timed Automata

Fast zone-based algorithms for reachability in pushdown timed automata

Timed Systems through the Lens of Logic